Mean value methods for solving the heat equation backwards in time

TL;DR

This paper presents an iterative mean value approach for solving the highly ill-posed inverse heat equation backwards in time, with convergence proofs and rates based on semi-group theory.

Contribution

It introduces a novel mean value iterative method for the inverse heat equation and provides rigorous convergence analysis and rates using semi-group theory.

Findings

Convergence of the method is proven mathematically.

Explicit convergence rates for residuals are established.

Convergence rates for iterates are derived under source conditions.

Abstract

We investigate an iterative mean value method for the inverse (and highly ill-posed) problem of solving the heat equation backwards in time. Semi-group theory is used to rewrite the solution of the inverse problem as the solution of a fixed point equation for an affine operator, with linear part satisfying special functional analytical properties. We give a convergence proof for the method and obtain convergence rates for the residual. Convergence rates for the iterates are also obtained under the so called source conditions.